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An ordering for groups of pure braids and fibre-type hyperplane arrangements. (English) Zbl 1047.20027

The braid group \(B_n\) admits a presentation with generators \(\sigma_1,\dots,\sigma_{n-1}\) and relations \(\sigma_i\sigma_j=\sigma_j\sigma_i\) if \(|i-j|>1\) for all \(i,j=1,\dots,n-1\) and \(\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\) for all \(i=1,\dots,n-2\). The pure braids form a normal subgroup \(P_n\) of index \(n!\). P. Dehornoy [J. Knot Theory Ramifications 4, No. 1, 33-79 (1995; Zbl 0873.20030] proved that the group \(B_n\) admits one sided orders but it is not two sided orderable (bi-orderable in the terminology used by the authors) if \(n\geq 3\). On the other hand, \(P_n\) admits two sided orders and the authors exhibit one such total order on \(P_n\) (and also of the fibre-type hyperplane arrangement groups) using the fact that such groups are residually torsion-free nilpotent. The paper contains structural description of \(P_n\) from several points of view.

MSC:

20F36 Braid groups; Artin groups
20F60 Ordered groups (group-theoretic aspects)
06F15 Ordered groups
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

Citations:

Zbl 0873.20030
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